Introducing viewpoints of mechanics into basic growth analysis - (XII) applying sign reversal of phase, space inversion and time reversal to wave functions with negative amplitude

This study was conducted to apply sign reversal of phase, space inversion and time reversal to wave functions with negative amplitude. The negative amplitude occurred when ± signs are given to each of three terms including amplitude when the square root was taken to solve a differential equation. Although a phase shift of π changed negative amplitude into positive amplitude, there were some mathematical operations that might be expected to avoid negative amplitude. The results obtained were as follows. Space inversion showed reversing the directions of all three-space coordinates of the wave. Time reversal showed the wave moving backward in time. The product of sign reversal of phase and space inversion or time reversal gave positive amplitude to the wave. Boldly speaking at the risk of making mistake, these mathematical operations suggested that the wave with positive amplitude and sign reversal of phase moved along reversal directions of space and backward in time. They were summarized by the transformation of coordinates from (ir, x, y, z, t, -W_o) to (-ir, -x, -y, -z, -t, W_o). It was suggested from the present study that the negative amplitude in wave functions was hypothetically avoided by applying sign reversal of phase, space inversion and time reversal.